Representation Varieties of Stacks and Trace Maps
Abstract
We define a derived stack ℛepn(X) which generalizes the assignment Aleadsto Repn(A) to an algebra of its derived GLn-representation variety of arXiv:1112.1449 from algebras A to perfect stacks X over characteristic 0 fields. In the case of a quasi-projective classical scheme X, we show that ℛepn(X) admits a subfunctor Quotn,fr𝒪Xn(X)⊂ ℛepn(X), which is in fact represented by a derived scheme almost of finite type. We further construct a Fourier-Mukai integral transform between the derived categories of quasi-coherent sheaves on X and Quotn,fr𝒪Xn(X) which induces a trace map at the level of Hochschild homology generalizing the trace morphism constructed in arXiv:1112.1449 (for finitely presented commutative algebras). We show that the subfunctor Quotn,fr𝒪Xn(X) is a derived enhancement of the framed locus of the Quot scheme of points and that this derived scheme is a GLn-torsor over the stack of coherent length n torsion sheaves. Hence, this stack is an analog of the derived character stack for quasi-projective schemes, and we show that for smooth, Calabi-Yau X, it inherits a shifted symplectic structure in the sense of arXiv:1111.3209 from the one constructed in arXiv:1812.11913 on the moduli stack of perfect complexes with proper support. This structure is shown to give a generalization of the classical symplectic structure on character varieties of surfaces of genus 1 constructed by Goldman \cite{gold}.
Related
BibTeX
@misc{erlikhman2026representation,
title = {{Representation Varieties of Stacks and Trace Maps}},
author = {Jacob Erlikhman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09633},
eprint = {2609.09633},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09633},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Representation Varieties of Stacks and Trace Maps
AU - Jacob Erlikhman
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09633
UR - https://arxiv.org/abs/2609.09633
AB - We define a derived stack ℛep_n(X) which generalizes the assignment Aleadsto Rep_n(A) to an algebra of its derived GL_n-representation variety of arXiv:1112.1449 from algebras A to perfect stacks X over characteristic 0 fields. In the case of a quasi-projective classical scheme X, we show that ℛep_n(X) admits a subfunctor Quot^n,fr_𝒪_X^n(X)⊂ ℛep_n(X), which is in fact represented by a derived scheme almost of finite type. We further construct a Fourier-Mukai integral transform between the derived categories of quasi-coherent sheaves on X and Quot^n,fr_𝒪_X^n(X) which induces a trace map at the level of Hochschild homology generalizing the trace morphism constructed in arXiv:1112.1449 (for finitely presented commutative algebras). We show that the subfunctor Quot^n,fr_𝒪_X^n(X) is a derived enhancement of the framed locus of the Quot scheme of points and that this derived scheme is a GL_n-torsor over the stack of coherent length n torsion sheaves. Hence, this stack is an analog of the derived character stack for quasi-projective schemes, and we show that for smooth, Calabi-Yau X, it inherits a shifted symplectic structure in the sense of arXiv:1111.3209 from the one constructed in arXiv:1812.11913 on the moduli stack of perfect complexes with proper support. This structure is shown to give a generalization of the classical symplectic structure on character varieties of surfaces of genus 1 constructed by Goldman \cite{gold}.
ER -